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Iterative Neural Autoregressive Distribution
Estimator (NADE-k)
Tapani Raiko
Aalto University
Li Yao
Universit´e de Montr´eal
KyungHyun Cho
Universit´e de Montr´eal
Yoshua Bengio
Universit´e de Montr´eal,
CIFAR Senior Fellow
Abstract
Training of the neural autoregressive density estimator (NADE) can be viewed as
doing one step of probabilistic inference on missing values in data. We propose
a new model that extends this inference scheme to multiple steps, arguing that
it is easier to learn to improve a reconstruction in k steps rather than to learn to
reconstruct in a single inference step. The proposed model is an unsupervised
building block for deep learning that combines the desirable properties of NADE
and multi-prediction training: (1) Its test likelihood can be computed analytically,
(2) it is easy to generate independent samples from it, and (3) it uses an inference
engine that is a superset of variational inference for Boltzmann machines. The
proposed NADE-k is competitive with the state-of-the-art in density estimation
on the two datasets tested.
1
Introduction
Traditional building blocks for deep learning have some unsatisfactory properties. Boltzmann machines are, for instance, difficult to train due to the intractability of computing the statistics of the
model distribution, which leads to the potentially high-variance MCMC estimators during training
(if there are many well-separated modes (Bengio et al., 2013)) and the computationally intractable
objective function. Autoencoders have a simpler objective function (e.g., denoising reconstruction
error (Vincent et al., 2010)), which can be used for model selection but not for the important choice
of the corruption function. On the other hand, this paper follows up on the Neural Autoregressive
Distribution Estimator (NADE, Larochelle and Murray, 2011), which specializes previous neural
auto-regressive density estimators (Bengio and Bengio, 2000) and was recently extended (Uria et al.,
2014) to deeper architectures. It is appealing because both the training criterion (just log-likelihood)
and its gradient can be computed tractably and used for model selection, and the model can be
trained by stochastic gradient descent with backpropagation. However, it has been observed that the
performance of NADE has still room for improvement.
The idea of using missing value imputation as a training criterion has appeared in three recent papers. This approach can be seen either as training an energy-based model to impute missing values
well (Brakel et al., 2013), as training a generative probabilistic model to maximize a generalized
pseudo-log-likelihood (Goodfellow et al., 2013), or as training a denoising autoencoder with a masking corruption function (Uria et al., 2014). Recent work on generative stochastic networks (GSNs),
which include denoising auto-encoders as special cases, justifies dependency networks (Heckerman et al., 2000) as well as generalized pseudo-log-likelihood (Goodfellow et al., 2013), but have
the disadvantage that sampling from the trained “stochastic fill-in” model requires a Markov chain
(repeatedly resampling some subset of the values given the others). In all these cases, learning
progresses by back-propagating the imputation (reconstruction) error through inference steps of the
model. This allows the model to better cope with a potentially imperfect inference algorithm. This
learning-to-cope was introduced recently in 2011 by Stoyanov et al. (2011) and Domke (2011).
1
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Figure 1: The choice of a structure for NADE-k is very flexible. The dark filled halves indicate that
a part of the input is observed and fixed to the observed values during the iterations. Left: Basic
structure corresponding to Equations (6–7) with n = 2 and k = 2. Middle: Depth added as in
NADE by Uria et al. (2014) with n = 3 and k = 2. Right: Depth added as in Multi-Prediction Deep
Boltzmann Machine by Goodfellow et al. (2013) with n = 2 and k = 3. The first two structures are
used in the experiments.
The NADE model involves an ordering over the components of the data vector. The core of the
model is the reconstruction of the next component given all the previous ones. In this paper we
reinterpret the reconstruction procedure as a single iteration in a variational inference algorithm,
and we propose a version where we use k iterations instead, inspired by (Goodfellow et al., 2013;
Brakel et al., 2013). We evaluate the proposed model on two datasets and show that it outperforms
the original NADE (Larochelle and Murray, 2011) as well as NADE trained with the order-agnostic
training algorithm (Uria et al., 2014).
2
Proposed Method: NADE-k
We propose a probabilistic model called NADE-k for D-dimensional binary data vectors x. We start
by defining pθ for imputing missing values using a fully factorial conditional distribution:
Y
pθ (xmis | xobs ) =
pθ (xi | xobs ),
(1)
i∈mis
where the subscripts mis and obs denote missing and observed components of x. From the conditional distribution pθ we compute the joint probability distribution over x given an ordering o (a
permutation of the integers from 1 to D) by
pθ (x | o) =
D
Y
pθ (xod | xo<d ),
(2)
d=1
where o<d stands for indices o1 . . . od−1 .
The model is trained to minimize the negative log-likelihood averaged over all possible orderings o
L(θ) = Eo∈D! [Ex∈data [− log pθ (x | o)]] .
(3)
using an unbiased, stochastic estimator of L(θ)
D
log pθ (xo≥d | xo<d )
(4)
D−d+1
by drawing o uniformly from all D! possible orderings and d uniformly from 1 . . . D (Uria et al.,
2014). Note that while the model definition in Eq. (2) is sequential in nature, the training criterion
(4) involves reconstruction of all the missing values in parallel. In this way, training does not involve
picking or following specific orders of indices.
ˆ
L(θ)
=−
In this paper, we define the conditional model pθ (xmis | xobs ) using a deep feedforward neural
network with nk layers, where we use n weight matrices k times. This can also be interpreted as
running k successive inference steps with an n-layer neural network.
The input to the network is
vh0i = m Ex∈data [x] + (1 − m) x
(5)
where m is a binary mask vector indicating missing components with 1, and is an elementwise multiplication. Ex∈data [x] is an empirical mean of the observations. For simplicity, we give
2
Figure 2: The inner working mechanism of NADE-k. The left most column shows the data vectors x,
the second column shows their masked version and the subsequent columns show the reconstructions
vh0i . . . vh10i (See Eq. (7)).
equations for a simple structure with n = 2. See Fig. 1 (left) for the illustration of this simple
structure.
In this case, the activations of the layers at the t-th step are
hhti = φ(Wvht−1i + c)
(6)
vhti = m σ(Vhhti + b) + (1 − m) x
(7)
where φ is an element-wise nonlinearity, σ is a logistic sigmoid function, and the iteration index t
runs from 1 to k. The conditional probabilities of the variables (see Eq. (1)) are read from the output
vhki as
hki
pθ (xi = 1 | xobs ) = vi .
(8)
Fig. 2 shows examples of how vhti evolves over iterations, with the trained model.
The parameters θ = {W, V, c, b} can be learned by stochastic gradient descent to minimize −L(θ)
ˆ
in Eq. (3), or its stochastic approximation −L(θ)
in Eq. (4), with the stochastic gradient computed
by back-propagation.
Once the parameters θ are learned, we can define a mixture model by using a uniform probability
over a set of orderings O. We can compute the probability of a given vector x as a mixture model
1 X
pmixt (x | θ, O) =
pθ (x | o)
(9)
|O|
o∈O
with Eq. (2). We can draw independent samples from the mixture by first drawing an ordering o and
then sequentially drawing each variable using xod ∼ pθ (xod | xo<d ). Furthermore, we can draw
samples from the conditional p(xmis | xobs ) easily by considering only orderings where the observed
indices appear before the missing ones.
Pretraining It is well known that training deep networks is difficult without pretraining, and in our
experiments, we train networks up to kn = 7 × 3 = 21 layers. When pretraining, we train the model
to produce good reconstructions vhti at each step t = 1 . . . k. More formally, in the pretraining
phase, we replace Equations (4) and (8) by
Lˆpre (θ) = −
k
Y hti
1X
D
log
pθ (xi | xo<d )
D − d + 1 k t=1
i∈o
(10)
≥d
hti
hti
pθ (xi = 1 | xobs ) = vi .
2.1
(11)
Related Methods and Approaches
Order-agnostic NADE The proposed method follows closely the order-agnostic version of
NADE (Uria et al., 2014), which may be considered as the special case of NADE-k with k = 1. On
the other hand, NADE-k can be seen as a deep NADE with some specific weight sharing (matrices
W and V are reused for different depths) and gating in the activations of some layers (See Equation
(7)).
3
Additionally, Uria et al. (2014) found it crucial to give the mask m as an auxiliary input to the
network, and initialized missing values to zero instead of the empirical mean (See Eq. (5)). Due to
these differences, we call their approach NADE-mask. One should note that NADE-mask has more
parameters due to using the mask as a separate input to the network, whereas NADE-k is roughly k
times more expensive to compute.
Probabilistic Inference Let us consider the task of missing value imputation in a probabilistic
latent variable model. We get the conditional probability of interest by marginalizing out the latent
variables from the posterior distribution:
Z
p(xmis | xobs ) =
p(h, xmis | xobs )dh.
(12)
h
Accessing the joint distribution p(h, xmis | xobs ) directly is often harder than alternatively updating
h and xmis based on the conditional distributions p(h | xmis , xobs ) and p(xmis | h).1 Variational
inference is one of the representative examples that exploit this.
In variational inference, a factorial distribution q(h, xmis ) = q(h)q(xmis ) is iteratively fitted to
p(h, xmis | xobs ) such that the KL-divergence between q and p
Z
p(h, xmis | xobs )
KL[q(h, xmis )||p(h, xmis | xobs )] = −
q(h, xmis ) log
dhdxmis
(13)
q(h, xmis )
h,xmis
is minimized. The algorithm alternates between updating q(h) and q(xmis ), while considering the
other one fixed.
As an example, let us consider a restricted Boltzmann machine (RBM) defined by
p(v, h) ∝ exp(b> v + c> h + h> Wv).
(14)
We can fit an approximate posterior distribution parameterized as q(vi = 1) = v¯i and q(hj = 1) =
¯ j to the true posterior distribution by iteratively computing
h
¯ ← σ(W¯
h
v + c)
(15)
¯ ← m σ(W> h + b) + (1 − m) v.
v
(16)
>
We notice the similarity to Eqs. (6)–(7): If we assume φ = σ and V = W , the inference in the
NADE-k is equivalent to performing k iterations of variational inference on an RBM for the missing
values (Peterson and Anderson, 1987). We can also get variational inference on a deep Boltzmann
machine (DBM) using the structure in Fig. 1 (right).
Multi-Prediction Deep Boltzmann Machine Goodfellow et al. (2013) and Brakel et al. (2013)
use backpropagation through variational inference steps to train a deep Boltzmann machine. This
is very similar to our work, except that they approach the problem from the view of maximizing
the generalized pseudo-likelihood (Huang and Ogata, 2002). Also, the deep Boltzmann machine
lacks the tractable probabilistic interpretation similar to NADE-k (See Eq. (2)) that would allow
to compute a probability or to generate independent samples without resorting to a Markov chain.
Also, our approach is somewhat more flexible in the choice of model structures, as can be seen in
Fig. 1. For instance, in the proposed NADE-k, encoding and decoding weights do not have to be
shared and any type of nonlinear activations, other than a logistic sigmoid function, can be used.
Product and Mixture of Experts One could ask what would happen if we would define an ensemble
likelihood along the line of the training criterion in Eq. (3). That is,
− log pprod (x | θ) ∝ Eo∈D! [− log p(x | θ, o)] .
(17)
Maximizing this ensemble likelihood directly will correspond to training a product-of-experts
model (Hinton, 2000). However, this requires us to evaluate the intractable normalization constant
during training as well as in the inference, making the model not tractable anymore.
On the other hand, we may consider using the log-probability of a sample under the mixture-ofexperts model as the training criterion
− log pmixt (x | θ) = − log Eo∈D! [p(x | θ, o)] .
(18)
This criterion resembles clustering, where individual models may specialize in only a fraction of the
data. In this case, however, the simple estimator such as in Eq. (4) would not be available.
1
We make a typical assumption that observations are mutually independent given the latent variables.
4
Model
NADE 1HL(fixed order)
NADE 1HL
NADE 2HL
NADE-mask 1HL
NADE-mask 2HL
NADE-mask 4HL
EoNADE-mask 1HL(128 Ords)
EoNADE-mask 2HL(128 Ords)
Log-Prob.
-88.86
-99.37
-95.33
-92.17
-89.17
-89.60
-87.71
-85.10
Model
RBM (500h, CD-25)
DBN (500h+2000h)
DARN (500h)
DARN (500h, adaNoise)
NADE-5 1HL
NADE-5 2HL
EoNADE-5 1HL(128 Ords)
EoNADE-5 2HL(128 Ords)
Log-Prob.
≈ -86.34
≈ -84.55
≈ -84.71
≈ -84.13
-90.02
-87.14
-86.23
-84.68
Table 1: Results obtained on MNIST using various models and number of hidden layers (1HL
or 2HL). “Ords” is short for “orderings”. These are the average log-probabilities of the test set.
EoNADE refers to the ensemble probability (See Eq. (9)). From here on, in all figures and tables we
use “HL” to denote the number of hidden layers and “h” for the number of hidden units.
3
Experiments
We study the proposed model with two datasets: binarized MNIST handwritten digits and Caltech
101 silhouettes.
We train NADE-k with one or two hidden layers (n = 2 and n = 3, see Fig. 1, left and middle)
with a hyperbolic tangent as the activation function φ(·). We use stochastic gradient descent on
the training set with a minibatch size fixed to 100. We use AdaDelta (Zeiler, 2012) to adaptively
choose a learning rate for each parameter update on-the-fly. We use the validation set for earlystopping and to select the hyperparameters. With the best model on the validation set, we report the
log-probability computed on the test set. We have made our implementation available2 .
3.1
MNIST
We closely followed the procedure used by Uria et al. (2014), including the split of the dataset into
50,000 training samples, 10,000 validation samples and 10,000 test samples. We used the same
version where the data has been binarized by sampling.
We used a fixed width of 500 units per hidden layer. The number of steps k was selected among
{1, 2, 4, 5, 7}. According to our preliminary experiments, we found that no separate regularization
was needed when using a single hidden layer, but in case of two
hidden layers, we used weight
decay with the regularization constant in the interval e−5 , e−2 . Each model was pretrained for
1000 epochs and fine-tuned for 1000 epochs in the case of one hidden layer and 2000 epochs in the
case of two.
For both NADE-k with one and two hidden layers, the validation performance was best with k = 5.
The regularization constant was chosen to be 0.00122 for the two-hidden-layer model.
Results We report in Table 1 the mean of the test log-probabilities averaged over randomly selected
orderings. We also show the experimental results by others from (Uria et al., 2014; Gregor et al.,
2014). We denote the model proposed in (Uria et al., 2014) as a NADE-mask.
From Table 1, it is clear that NADE-k outperforms the corresponding NADE-mask both with the
individual orderings and ensembles over orderings using both 1 or 2 hidden layers. NADE-k with
two hidden layers achieved the generative performance comparable to that of the deep belief network
(DBN) with two hidden layers.
Fig. 3 shows training curves for some of the models. We can see that the NADE-1 does not perform
as well as NADE-mask. This confirms that in the case of k = 1, the auxiliary mask input is indeed
useful. Also, we can note that the performance of NADE-5 is still improving at the end of the
preallocated 2000 epochs, further suggesting that it may be possible to obtain a better performance
simply by training longer.
2
[email protected]:yaoli/nade k.git
5
120
−90
115
training cost
110
105
100
95
90
−94
end of pretrain
−96
NADE-mask 1HL
NADE-5 1HL
NADE-1 1HL
−98
end of pretrain
85
80
−92
testset log-probability
NADE-mask 1HL
NADE-5 1HL
NADE-1 1HL
0
500
1000
−100
200
1500
training epochs
400
600
(a)
800
1000
1200
1400
training epochs
1600
1800
2000
(b)
Figure 3: NADE-k with k steps of variational inference helps to reduce the training cost (a) and to
generalize better (b). NADE-mask performs better than NADE-1 without masks both in training and
test.
−87
−85
−90
−89
−90
−91
−92
−93
NADE-k 1HL
NADE-k 2HL
NADE-mask 1HL
NADE-mask 2HL
−94
−95
−96
1
2
4
5
trained with k steps of iterations
testset log-probability
testset log-probability
−88
−95
−100
−105
−110
−115
7
(a)
NADE-5 2HL
NADE-mask 2HL
0
5
10
15
perform k steps of iterations at test time
20
(b)
Figure 4: (a) The generalization performance of different NADE-k models trained with different k.
(b) The generalization performance of NADE-5 2h, trained with k=5, but with various k in test time.
Fig. 4 (a) shows the effect of the number of iterations k during training. Already with k = 2, we can
see that the NADE-k outperforms its corresponding NADE-mask. The performance increases until
k = 5. We believe the worse performance of k = 7 is due to the well known training difficulty of a
deep neural network, considering that NADE-7 with two hidden layers effectively is a deep neural
network with 21 layers.
At inference time, we found that it is important to use the exact k that one used to train the model.
As can be seen from Fig. 4 (b), the assigned probability increases up to the k, but starts decreasing
as the number of iterations goes over the k. 3
3.1.1
Qualitative Analysis
In Fig. 2, we present how each iteration t = 1 . . . k improves the corrupted input (vhti from Eq. (5)).
We also investigate what happens with test-time k being larger than the training k = 5. We can see
that in all cases, the iteration – which is a fixed point update – seems to converge to a point that is
in most cases close to the ground-truth sample. Fig. 4 (b) shows however that the generalization
performance drops after k = 5 when training with k = 5. From Fig. 2, we can see that the
reconstruction continues to be sharper even after k = 5, which seems to be the underlying reason
for this phenomenon.
3
In the future, one could explore possibilities for helping better converge beyond step k, for instance by
using costs based on reconstructions at k − 1 and k even in the fine-tuning phase.
6
(b) Caltech-101 Silhouettes
(a) MNIST
Figure 5: Samples generated from NADE-k trained on (a) MNIST and (b) Caltech-101 Silhouettes.
(b)
(a)
Figure 6: Filters learned from NADE-5 2HL. (a) A random subset of the encodering filters. (b) A
random subset of the decoding filters.
From the samples generated from the trained NADE-5 with two hidden layers shown in Fig. 5 (a),
we can see that the model is able to generate digits. Furthermore, the filters learned by the model
show that it has learned parts of digits such as pen strokes (See Fig. 6).
3.1.2
Variability over Orderings
In Section 2, we argued that we can perform any inference task p(xmis | xobs ) easily and efficiently
by restricting the set of orderings O in Eq. (9) to ones where xobs is before xmis . For this to work
well, we should investigate how much the different orderings vary.
To measure the variability over orderings, we computed the variance of log p(x | o) for 128 randomly chosen orderings o with the trained NADE-k’s and NADE-mask with a single hidden layer.
For comparison, we computed the variance of log p(x | o) over the 10,000 test samples.
p
p
log p(x | o)
Eo,x [·]
Ex Varo [·]
Eo Varx [·]
Table 2: The variance of
NADE-mask 1HL -92.17
3.5
23.5
log p(x | o) over orderings o
NADE-5 1HL
-90.02
3.1
24.2
and over test samples x.
NADE-5 2HL
-87.14
2.4
22.7
In Table 2, the variability over the orderings is clearly much smaller than that over the samples.
Furthermore, the variability over orderings tends to decrease with the better models.
3.2
Caltech-101 silhouettes
We also evaluate the proposed NADE-k on Caltech-101 Silhouettes (Marlin et al., 2010), using
the standard split of 4100 training samples, 2264 validation samples and 2307 test samples. We
demonstrate the advantage of NADE-k compared with NADE-mask under the constraint that they
have a matching number of parameters. In particular, we compare NADE-k with 1000 hidden
units with NADE-mask with 670 hiddens. We also compare NADE-k with 4000 hidden units with
NADE-mask with 2670 hiddens.
We optimized the hyper-parameter k ∈ {1, 2, . . . , 10} in the case of NADE-k. In both NADE-k
and NADE-mask, we experimented without regularizations, with weight decays, or with dropout.
Unlike the previous experiments, we did not use the pretraining scheme (See Eq. (10)).
7
Table 3: Average log-probabilities of test samples of Caltech-101 Silhouettes. (?) The results are
from Cho et al. (2013). The terms in the parenthesis indicate the number of hidden units, the total
number of parameters (M for million), and the L2 regularization coefficient. NADE-mask 670h
achieves the best performance without any regularizations.
Model
RBM?
(2000h, 1.57M)
NADE-mask
(670h, 1.58M)
NADE-2
(1000h, 1.57M, L2=0.0054)
Test LL
-108.98
Model
RBM ?
(4000h, 3.14M)
NADE-mask
(2670h, 6.28M, L2=0.00106)
NADE-5
(4000h, 6.28M, L2=0.0068)
-112.51
-108.81
Test LL
-107.78
-110.95
-107.28
As we can see from Table 3, NADE-k outperforms the NADE-mask regardless of the number of
parameters. In addition, NADE-2 with 1000 hidden units matches the performance of an RBM with
the same number of parameters. Futhermore, NADE-5 has outperformed the previous best result
obtained with the RBMs in (Cho et al., 2013), achieving the state-of-art result on this dataset. We
can see from the samples generated by the NADE-k shown in Fig. 5 (b) that the model has learned
the data well.
4
Conclusions and Discussion
In this paper, we proposed a model called iterative neural autoregressive distribution estimator
(NADE-k) that extends the conventional neural autoregressive distribution estimator (NADE) and its
order-agnostic training procedure. The proposed NADE-k maintains the tractability of the original
NADE while we showed that it outperforms the original NADE as well as similar, but intractable
generative models such as restricted Boltzmann machines and deep belief networks.
The proposed extension is inspired from the variational inference in probabilistic models such as
restricted Boltzmann machines (RBM) and deep Boltzmann machines (DBM). Just like an iterative
mean-field approximation in Boltzmann machines, the proposed NADE-k performs multiple iterations through hidden layers and a visible layer to infer the probability of the missing value, unlike
the original NADE which performs the inference of a missing value in a single iteration through
hidden layers.
Our empirical results show that this approach of multiple iterations improves the performance of
a model that has the same number of parameters, compared to performing a single iteration. This
suggests that the inference method has significant effect on the efficiency of utilizing the model
parameters. Also, we were able to observe that the generative performance of NADE can come
close to more sophisticated models such as deep belief networks in our approach.
In the future, more in-depth analysis of the proposed NADE-k is needed. For instance, a relationship between NADE-k and the related models such as the RBM need to be both theoretically and
empirically studied. The computational speed of the method could be improved both in training (by
using better optimization algorithms. See, e.g., (Pascanu and Bengio, 2014)) and in testing (e.g. by
handling the components in chunks rather than fully sequentially). The computational efficiency of
sampling for NADE-k can be further improved based on the recent work of Yao et al. (2014) where
an annealed Markov chain may be used to efficiently generate samples from the trained ensemble.
Another promising idea to improve the model performance further is to let the model adjust its own
confidence based on d. For instance, in the top right corner of Fig. 2, we see a case with lots of missing values values (low d), where the model is too confident about the reconstructed digit 8 instead
of the correct digit 2.
Acknowledgements
The authors would like to acknowledge the support of NSERC, Calcul Qu´ebec, Compute Canada,
the Canada Research Chair and CIFAR, and developers of Theano (Bergstra et al., 2010; Bastien
et al., 2012).
8
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